The Proof of One

[Epsilon=One]

Some musings:

I would think that number theory is on shaky ground until the value of “One” can be established such that it is a constant throughout any system.

It also seems logical that the numbers and their relationship to each other must be established from natural origins for number theory to be more than a mechanical contrivance.

Such should then directly relate the Fibonacci sequence to Pi, the Golden Ratio, and sinusoidal and elliptical curves; and thus, demonstrate why both the FS and the Golden Ratio are ubiquitous throughout nature. A term to describe this relationship could be referred to as the Natural function (http://www.physicsmathforums.com/showthread.php?t=121).

Mathematics does not
......explain Nature;
Nature explains
......mathematics.
All mathematics is
......a function of Nature;
thus,
......its sublime poetry . . .

[JasonRox]

What do you mean by the definition of the number 1?

It seems pretty well established through the Peano Axioms.

[Epsilon=One]

What do you mean by the definition of the number 1?

It seems pretty well established through the Peano Axioms.All mathematics is unprovable until "One" can be proved from within any system. The Peano Axioms do not assign a value to "One"; they merely state that it exists. Other numbers are functions of "One."

All mathematical numbers represent values that are functions of an arbitrary value for "One"; said arbitrary value must be proven.

The solution depends upon defining "One" in terms of the given system of numbers without the definition being circular. In essence, this is what Kurt Gödel (http://www.usna.edu/Users/math/meh/godel.html) said was most unlikely, or metaphysical, with his Incompleteness Theory. (http://users.ox.ac.uk/~jrlucas/Godel/simplex.html)

Pulsoid Theory's (http://www.physicsmathforums.com/showthread.php?t=128) Natural source of numbers (http://www.physicsmathforums.com/showthread.php?t=141) and their mathematical manipulations are related to the Proof of One.

See: "Universal Proof of One." (http://www.physicsmathforums.com/showthread.php?t=165)

[phyti]

The standard definition of a prime integer is: an integer divisible only by itself and 1.
Since all integers are divisible by 1, this cannot be a qualification.
The integer 1 is only divisible by itself but is excluded to preserve the uniqueness feature of the fundamental theorem of arithmetic
The fundamental theorem of arithmetic states: all integers>1 are the unique product of primes, even though prime integers have no factors (a product requires 2 factors). A single prime is not a product unless paired with 1, but this would allow multiple one's, which violates the unique product aspect of the theorem.
These two 'fundamental' examples alone show the need for consistent definitions without arbitrary restrictions and fudging to make things work.
An alternate prime definition is: a prime integer is a multiple of 1 only.
The integer 1 is excluded by definition because it is not a multiple.
The fundamental theorem of arithmetic could be redefined as:
all composite integers are the unique product of primes.
These defintions require a different concept in which the integers contain
three mutually exclusive classes of objects, {0, 1, M} which eliminates division
by zero, and the definiton of identity elements.
The primes stand alone and 1 is not a factor in any integer because it is not a prime. If any interest I have a paper on a simple set development for the integers. phyti

[Epsilon=One]

The standard definition of a prime integer is: an integer divisible only by itself and 1.
Since all integers are divisible by 1, this cannot be a qualification. Your definition is not exactly accurate even for the conventional definitions of prime numbers. More better: An integer not divisible by any integer factors except itself AND one. Your interpretation of being simply divisible by one would truly, as you point out, be wrong. You are very perceptive concerning the problems that few can see; however, I question your grasp of where the answers lie.

You are quite correct in asserting that the many definitions of “primes” are quite contrived and in many ways arbitrary. This fuzziness of definitions can be extended to the entire concept of numbers without too much effort.

There is only one definition of primes that should be considered as a Standard Model from which all other sets/definitions are derived as special cases. Any other beginning definition would be “incomplete.” The conventional definition, no matter how applied, is not actually even a single set of numbers.

The source of numbers and arithmetic is Nature where “primes” are very integral to the most seminal constructs . . . without "primes" there is no structure for seminal quanta. When Natural “primes” are understood all the mystery of the uniformity of primes vanishes. The distribution of Natural "primes" is uniform and with simple algebra all "primes" can be mapped to any uniform set of positive integers.

Nature does not consider one or two as “prime.”

I would be interested in your paper concerning the development of the integers; though, if this post flows from it, I don’t believe, as you assert, that you have an understanding of the origin of numbers; particularly, the status of One, Zero, the common denominator concept, and prime numbers. Have you considered an answer to Gödel’s Incompleteness Theorem?

[OfficeShredder]

Let zero be defined as the quantity of elements in the null set (we can all agree the null set exists, right? Because if it didn't, that would be weird).

Then take a new set, named A. A is the set of all subsets of zero. That is:

A = {null}.

1 is the quantity of elements in A.

Let B be the set of all subsets of A.

B = {null, {null}}.

2 is the quantity of elements in B.

etc. etc.

[Epsilon=One]

Let zero be defined as the quantity of elements in the null set (we can all agree the null set exists, right? Because if it didn't, that would be weird).

Then take a new set, named A. A is the set of all subsets of zero. That is:

A = {null}.

1 is the quantity of elements in A.

Let B be the set of all subsets of A.

B = {null, {null}}.

2 is the quantity of elements in B.

etc. etc.The above comments hardly solve any of the problems concerning "One" or the Natural source of numbers.

To begin: sets are contrivances.

Zero can not be so easily defined; because, zero has several distinct connotations that must be defined before it can be associated with any set.

It's rather hard to understand the existence of the "null set," whether that be "weird" or not, until you define existence. For many, symbols are not existence, but merely placeholders for the "real" thing . . . that usually exists.

Your use of "1" as the "quantity of elements in A" would hardly satisfy the requirements of Gödel's Incompleteness.

[OfficeShredder]

To begin: sets are contrivances.

That depends on how you look at things. While obviously we define a set in a mathematical sense (meaning it's contrived for us), there's no reason for you to not consider the possibility that the single most fundamental building block in the universe is the concept of a collection. Atoms are a collection of protons, neutrons, and electrons, which are collections of quarks, etc. There's no reason to pass sets off as a contrived idea; similiarly, any concept in any field could be so passed off.

Zero can not be so easily defined; because, zero has several distinct connotations that must be defined before it can be associated with any [set.

You can define the quantity of zero to be the number of elements in the null set. The rest of the properties can be followed up on just like they were over the course of history.

It's rather hard to understand the existence of the "null set," whether that be "weird" or not, until you define existence. For many, symbols are not existence, but merely placeholders for the "real" thing . . . that usually exists.

The great thing about the null set is that it exists even if nothing else does (in fact, you could call everything the null set then).

Your use of "1" as the "quantity of elements in A" would hardly satisfy the requirements of Gödel's Incompleteness.

I'm not quite sure what you're looking for anymore...


I would think that number theory is on shaky ground until the value of “One” can be established such that it is a constant throughout any system.

It also seems logical that the numbers and their relationship to each other must be established from natural origins for number theory to be more than a mechanical contrivance.

There is a single particular value of 1 using the set addition definition throughout any system. Furthermore, ignoring your qualm about sets not being "real" enough for you, it's a natural relationship from one number to the next (particularly from 0 to 1), and not a simple mechanical contrivance. You can even add and multiply the set values of natural numbers, and it works out great.

[phyti]

To form knowledge the mind;
perceives reality,
forms concepts to model reality,
predicts reality from these concepts,
retains the concepts as knowledge when prediction matches reality.
modifies the concepts as needed.
Knowledge is a set of concepts used as a reference for understanding.
By definition knowledge is always incomplete because all reality is never perceived.For simplicity, a concept is defined within a context that excludes other concepts.
Other concepts may not be relevant to the purpose.
There may be relevant concepts that have not been discovered.
An approximate definition may be sufficient for the purpose.
Definition is a relative referencing process.
A definition is expressed in terms of other definitions.
This process can be circular or incomplete.
Forming knowledge is a continuous process of refinement.
Knowledge is only as good as its definitions
In summary we know reality indirectly through images.
The preceding forms a basis for the following.
If knowledge is incomplete then in principle all systems,
including those analyzed by Goedel are incomplete.
History supports this view by the constant revision of theories and methodologies.
The concept of number was created from a need to measure things.
To measure is to compare an object to a reference object to evaluate a property
A property is an attribute/quality of an object that identifies it from other objects [color, texture, shape, hardness]
A unit is a concept defined within a context as simple, basic, indivisible [i.e. anything can be defined as a unit]
The simplest natural number set is the fingers, used for counting.
One is a relative concept depending on perspective. A person has one car but to the mechanic it is many parts.
The occurrence in the natural world, of spirals, Fibonacci sequences, etc. to me indicate design and order,
something that cannot be explained by random variation.
0 = nothing: literally "no thing", a condition or state of emptiness
Zero and one (nothing and something), are mutually exclusive concepts.
On that basis alone, how can 0 be treated as a typical number? A place holder, yes, a different type, yes.
What is one of the biggest bugs of all programmers,... division by zero.
Ask yourself, why can't I divide by zero. Division works for all other numbers so the operation can't be at fault,
that leaves zero as the problem.
My suggestion is, stop reciting the axioms like a mantra, get a good dictionary, and think outside the box.
Discovery is wonderful!

[OfficeShredder]

On that basis alone, how can 0 be treated as a typical number? A place holder, yes, a different type, yes.

It's not... it's the only number that's neither positive nor negative. This is important.

What is one of the biggest bugs of all programmers,... division by zero.
Ask yourself, why can't I divide by zero. Division works for all other numbers so the operation can't be at fault,

Right off the bat, ignoring convergence issues, the lack of it being positive or negative means you don't know whether the quotient is positive or negative. So it makes an intuitive sense that you can't divide by zero.

[Epsilon=One]

A unit is a concept defined within a context as simple, basic, indivisible [i.e. anything can be defined as a unit]Yes. But there is a fundamental unit within Nature. I refer to it as the Conceptual Unit that establishes the definition of fundamental, intrinsic time (FIT). FIT establishes the harmony that results in the resonances that are commonly known as light and matter.

The Conceptual Unit is heuristically demonstrated by the Elliptical Constant.

The simplest natural number set is the fingers, used for counting.Not even close. I know you weren't thinking too much here. Whatever the answer is, it must at least be binary.

One is a relative concept depending on perspective.Qualifiedly, Yes; but, correctly: No. If this were so, Kurt Gödel would be correct.

The occurrence in the natural world, of spirals, Fibonacci sequences, etc. to me indicate design and order, something that cannot be explained by random variation.Right on!!! And, that order begins with utmost simplicity.

0 = nothing: literally "no thing", a condition or state of emptiness
Zero and one (nothing and something), are mutually exclusive concepts.
On that basis alone, how can 0 be treated as a typical number? A place holder, yes, a different type, yes.
What is one of the biggest bugs of all programmers,... division by zero.
Ask yourself, why can't I divide by zero. Division works for all other numbers so the operation can't be at fault,
that leaves zero as the problem.
My suggestion is, stop reciting the axioms like a mantra, get a good dictionary, and think outside the box.
Discovery is wonderful!A little incomplete and clumsy . . . but, YOU'VE MADE MY DAY.

[Epsilon=One]

That depends on how you look at things. While obviously we define a set in a mathematical sense (meaning it's contrived for us), there's no reason for you to not consider the possibility that the single most fundamental building block in the universe is the concept of a collection. Atoms are a collection of protons, neutrons, and electrons, which are collections of quarks, etc. There's no reason to pass sets off as a contrived idea; similiarly, any concept in any field could be so passed off.Your argument is somewhat tautological. I won’t bother with its aptness.

You apparently see no difference in the value of concepts; or, the different importance of some disciplines from others when trying to ascertain Natural truths.

I can not agree with the triviality that you assign to definitions and their connotations. Before mathematicians take so dearly to their symbols I would suggest perusing Korzybski’s “Science and Sanity” (General Semantics).

You can define the quantity of zero to be the number of elements in the null set. The rest of the properties can be followed up on just like they were over the course of history.Reliance upon sets when discussing the singularity qualities of “One” and “zero” is much like expecting the word “cup” to hold water. History is a poor source for those seeking truth.

The great thing about the null set is that it exists even if nothing else does (in fact, you could call everything the null set then).Where from within the realm of metaphysics do your concepts of existence drift from?

There is a single particular value of 1 using the set addition definition throughout any system.Yes, and therein, with its selection lies the problem that Gödel exposed and thereby shredded most of early twentieth century philosophy. And, to this day no mathematician has disputed the relevance. If the system sets the value (as one might think from your statement) without relying upon an outside source; then, your sets have made “Incompleteness” moot and should acquire much fame for their implementer.

Furthermore, ignoring your qualm about sets not being "real" enough for you, it's a natural relationship from one number to the next (particularly from 0 to 1), and not a simple mechanical contrivance.You are using “0” and “1” as placeholders, which I consider mechanical. As such they could represent any two numbers. “One” and “Zero” are special and closely related with the infinite and infinitesimal, which are unlike any other number in any scale. The results of multiplication and division with each are unlike any other number.

Other numbers that are equidistant from zero and one on a scale (such as:
-2,+3; -7,+8, etc.) return the same value from the Natural function (NF),
(x² – x). Thus, the NF relates zero and one . . . unlike any other two numbers.

The entire subject is too long, tortuous, and subtle to be properly treated here. Perhaps, you might settle for discussing one small portion at a time?

You can even add and multiply the set values of natural numbers, and it works out great. You make the common mistake of many mathematicians. You rely upon symbols and axioms for a starting point without a concern for any fundamental proof that is consistent with a Natural source.

The Natural importance is not what occurs after the postulate; it is before the postulate where truth is found.

If you are correct concerning “One” and its manipulations you will put metaphysics on a firm footing much as have pomo, theoretical physicists by not beginning with the simplest fundamentals; also, I would think you would merit some of the most prestigious awards in mathematics.

[phyti]

You are looking for an objective 'one', independent of humanity.
My perspective is for a subjective 'one', and within the limitations of
human knowledge. I don't see any innate ability of the mind to understand
the reality of the universe, thus the reference to knowledge by images/concepts. Reasoning in any form, science, logic, etc. cannot explain
the fundamental substance, i.e. what something is, only how it behaves.
The mind is not omnipotent. My argument concerning the numbers o and 1, is a disagreement with the set theory gurus.
Division by zero was used because it is at the elementary level of number theory. If the base is faulty, how can you build on it successfully?
It seems we might share a common concern that sometimes, academia
exists for the sake of academia.
Glad you got something out of the post.

[Epsilon=One]

You are looking for an objective 'one', independent of humanity.Yes. Very few understand the difference.

I don't see any innate ability of the mind to understand the reality of the universe, thus the reference to knowledge by images/concepts.It is difficult not to think as you do; though, I am more optimistic than you seem to be. I consider the mind alone (well, maybe with pencil and paper) a far more powerful tool than the likes of the Hubble Space Telescope (HST) and the Large Hadron Collider (LHC) for eventually understanding "the reality of the universe." It is nice that the elite have such expensive tools (playthings), if only to convince them, from time-to-time, that they don’t understand either the fundamentals or the “big picture” of the environment in which we find ourselves.

I have no argument with your reference to images and concepts; just the opposite. Particularly, I liked, “Knowledge is only as good as its definitions.” and ”…we know reality indirectly through images.”

Reasoning in any form, science, logic, etc. cannot explain the fundamental substance, i.e. what something is, only how it behaves.Possibly so; however, I will not accept that there cannot be a rationalization as refined as required.

The mind is not omnipotent.I agree; however, I will never underestimate its potential!

My argument concerning the numbers o and 1, is a disagreement with the set theory gurus.You are on firm ground. Hold your position. It is difficult, but not impossible, to occasionally make some headway towards truth with those that set their own axioms.

Division by zero was used because it is at the elementary level of number theory. If the base is faulty, how can you build on it successfully?A point that seems to allude world class theoretical physicists and hubristic mathematicians.

It seems we might share a common concern that sometimes, academia
exists for the sake of academia.You have noted the weakest link in academia’s search for truth.

Glad you got something out of the post.What’s important is that others may . . .

[OfficeShredder]

Other numbers that are equidistant from zero and one on a scale (such as:
-2,+3; -7,+8, etc.) return the same value from the Natural function (NF),
(x² – x). Thus, the NF relates zero and one . . . unlike any other two numbers.

The entire subject is too long, tortuous, and subtle to be properly treated here. Perhaps, you might settle for discussing one small portion at a time?

In that case I'd like to start here. I'm not sure how you justify the NF singling out zero and one..... it also returns the same value for any two numbers equidistant from -2 and 3. While you could certainly consider zero and one as the "starting point", given that they are the numbers closest together, it's possibly just coincidence. Consider that if two numbers, x and y, yield the same value for the natural function:

x(x-1)=y(y-1)

then the first natural (no pun intended) step would be to say:

x=y-1
y=x-1

Of course, this yields 2=0 (who knows? Maybe it's true.... :cool: ). From there, the test of

x=-(y-1)
y=-(x-1)

yields two equations which are identical (if re-arranged). So essentially, you're looking for a set of numbers:

x=1-y

That (0,1) happens to be a solution yields no significance to me over any other solutions....

[Epsilon=One]

In that case I'd like to start here. I'm not sure how you justify the NF singling out zero and one..... it also returns the same value for any two numbers equidistant from -2 and 3. While you could certainly consider zero and one as the "starting point", given that they are the numbers closest together, it's possibly just coincidence.You are making my point. Any other numbers that are equidistant from -2 and 3 must also be equidistant from zero and one. This can only be said for zero and one despite the unending amount of other numbers.

I could ride with your "starting point"; however, there is far more depth and subtlety than just coincidence; which coincidence is not quite the situation anyway.

What I neglected was the source of the NF. The Pulse plus the NF equal the vector for any ellipse when the difference between the hypotenuse and focal length (wave) is the Elliptical Constant (One). Depending on whether the ellipse is acute or obtuse the NF is either the perigee or soliton (half the wave). For an equilateral ellipse (wave and vector are equal) there is no difference.

Thus, the NF is an extension or corallary of the Elliptical Constant, which most assurdedly must rank as one of the most important constants in mathematics . . . as all numbers are derived from it. Plus it yields the only known definition of fundamental "time."

I restate that, zero and one are unique among the numbers in that numbers equidistant from them return the same value for the NF. This is an important aspect of why their is no naturally occurring antimatter in the Universe.

Actually, zero is not a number; it indicates the absence of a number (value). And, one is a common denominator for the numbers of a system (set). Mathematics, which numbers are a part of, as with everything else that exists, must be derived from Nature. I believe you stated something to the effect that Nature is groups of things, collections, sets, etc.; try as I might, zero and one just do not seem to connote groups . . .

Somewhere your equations have gone wrong in evaluating the situation. If you choose any other adjacent numbers, say 7 and 8, then 4 and 11 are equidistant. The NF for 4 equals 12 and the NF for 11 equals 110; which is quite unlike starting with zero and one rather than 7 and 8.

Maybe you can let me know why we are miscommunicating???

That (0,1) happens to be a solution yields no significance to me over any other solutions....I would agree if there was any other solution that returned identical values.

I don't expect this post to clear the matter in itself; though, hopefully, it will help pinpoint our miscommunication.

Keep carping.